Asymptotic analysis of a mixed boundary value problem in a thick multi-level junction (Q2467895)

The authors are concerned with the asymptotic expansions as \(\varepsilon \rightarrow0\) for the solution to a mixed boundary value problem for the Poisson equation on a two-level junction \(\Omega_{\varepsilon}.\) A thick multi-level junction is the union of a domain \(\Omega_{0}\) and a large number \(N\) of thin domains with variable thickness of order \(\varepsilon=O(N^{-1}).\) Thin domains are divided into a finite number of levels depending on their length; thin domains at different levels are \(\varepsilon\)-periodically alternated along a certain manifold in the joint zone. The Poisson equation is considered subject to Robin boundary conditions on the boundaries of thin rods, periodic conditions on the vertical sides of \(\Omega_{0},\) and the Neumann condition on the rest of the boundary \(\partial\Omega_{\varepsilon}.\) Using the method of matching asymptotic expansions, the authors construct an asymptotic expansion for the unique weak solution in an anisotropic Sobolev vector space \(H^{1}(\Omega_{\varepsilon})\) and obtain error estimates.